Iterative method

An iterative method attempts to solve a problem, for example an equation or system of equations, by finding successive approximations to the solution.

In the case of a linear system the two main classes are the stationary iterative methods, and the more general Krylov subspace methods.

Table of contents

1 Stationary iterative methods
2 Krylov subspace methods
3 Convergence
4 Preconditioners
5 History
6 External links

Stationary iterative methods

Stationary iterative methods solve a system with an operator approximating the original one; and based on a measurement of the error (the residual) form a correction equation for which this process is repeated. While these methods are simple to derive, implement, and analyse, convergence is only guaranteed for a limited class of matrices.

Krylov subspace methods

Krylov subspace methods form an orthogonal basis of the sequence of successive matrix powers times the initial residual (the Krylov sequence). The approximations to the solution are then formed by minimizing the residual over the subspace formed. The prototypical method in this class is the conjugate gradient method.

Convergence

Since these methods form a basis, it is evident that the method converges in N iterations, where N is the system size. However, in the presence of rounding errors this statement does not hold; moreover, in practice N can be very large, and the iterative process reaches sufficient accuracy already far earlier. The analysis of these methods is hard, depending on a complicated function of the spectrum of the operator.

Preconditioners

The approximating operator that appears in stationary Iterative methods can also be incorporated in Krylov subspace methods (alternatively, preconditioned Krylov methods can be considered as accelerations of stationary iterative methods), where they become transformations of the original operator to a presumably better conditioned one. The construction of preconditioners is a large research area.

History

1671,1690,1736 The Newton-Raphson method explicitly iterates a formula to arrive at a solution.

Also, the iterative method appeared in a letter of Gauss to a student of his. He proposed solving a 4-by-4 system of equations by repeatedly solving the component in which the residual was largest.

The theory of Stationary Iterative methods was solidly established with the work of D.M. Young starting the 1950s. The Conjugate Gradient method was also invented in the 1950s, with independent developments by Cornelius Lanczos and David Hestenes and Stiefel, but its nature and applicability was minunderstood at the time. Only in the 1970s was it realized that conjugacy based methods work very well for partial differential equations, especially of elliptic type.

In the 1900s, Emil Post, Alonzo Church and others published the relationship of iteration to recursion. For example, the Scheme programming language expresses iteration with tail-recursion.

External links

Templates for the Solution of Linear Systems

The Iterative method is one of the software development processes, for example in Barry Boehm's 1980s spiral model, which influenced extreme programming practices in the 2000s. See iterative development for the detail.